Multicore Quantum Computing

Any architecture for practical quantum computing must be scalable. An attractive approach is to create multiple cores, computing regions of fixed size that are well spaced but interlinked with communication channels. This exploded architecture can relax the demands associated with a single monolithic device: the complexity of control, cooling and power infrastructure, as well as the difficulties of crosstalk suppression and near-perfect component yield. Here we explore interlinked multicore architectures through analytic and numerical modeling. While elements of our analysis are relevant to diverse platforms, our focus is on semiconductor electron spin systems in which numerous cores may exist on a single chip within a single fridge. We model shuttling and microwave-based interlinks and estimate the achievable fidelities, finding values that are encouraging but markedly inferior to intracore operations. We therefore introduce optimized entanglement purification to enable high-fidelity communication, finding that $99.5\mathrm{\%}$ is a very realistic goal. We then assess the prospects for quantum advantage using such devices in the noisy intermediate-scale quantum era and beyond: we simulate recently proposed exponentially powerful error mitigation schemes in the multicore environment and conclude that these techniques impressively suppress imperfections in both the inter- and intracore operations.

Figure 1

(a) Macroscopically separate quantum cores (core 1, core 2, etc.) are linked via purified Bell pairs (wavy lines). Such an architecture is well suited for implementing the ESD-VD error mitigation technique [26, 27]: multiple quantum cores perform the same quantum computation on $N$ qubits and these copies are used to verify each other via a derangement operation. This exponentially reduces the impact of imperfections in both the local operations and in the Bell pairs. The derangement operation is implemented by teleporting individual qubits into a buffer qubit (magenta dot) in one of the cores, thus consuming overall $N$ bell pairs. In this work we consider distributing Bell pairs via a cavity (b) and by shuttling (c) (not to scale). (b) Schematic of two spin registers connected via a superconducting cavity. By hybridizing the spin and charge degrees of freedom of electrons in DQDs, we can couple distant qubits through microwave photons (red squiggles) in a cavity. Charge measurements are used to speed up the process. A successful measurement outcome will lead to a raw Bell pair, whose quality can be improved through distillation. (c) Schematic of two spin registers connected via a quantum dot shuttling channel. A Bell pair consisting of two electrons (small orange dots) is created locally via high-fidelity local gates; one of the electrons is then shuttled through the few micron-length chain on a 100 ns timescale—this chain does not use micromagnets given we need not control the spin degree of freedom during shuttling.

Figure 2

Sketch of a single electron confined in a DQD. The DQD is generated by gates on top of a material heterostructure (a $\mathrm{Si}$/$\mathrm{Si}\mathrm{Ge}$ heterostructure for instance). Cobalt micromagnets are placed above the DQD to generate the transverse magnetic field $b_x$ that will hybridize the charge and spin degrees of freedom. By lifting the spin degeneracy, the parallel magnetic field $(0,0,B{)}^T$ allows us to define a qubit on the electron’s spin.

Figure 3

(a) Evolution of the probabilities of measuring the charges at different locations. (b) Evolution of the concurrence between the distant spins. Both plots are computed using the optimized evolution parameters. The optimal initial state is given by $|\psi ⟩\approx [(|L⟩-|R⟩)|\uparrow ⟩/\sqrt{2}]\otimes [(|L⟩-|R⟩)|\uparrow ⟩/\sqrt{2}]\otimes |0⟩$ with the first two terms describing the state of the first and second electrons, respectively, and the last one the state of the cavity. We account for potential difficulties to stop the interaction exactly at the maximal probability: we average the final density matrix over different stopping times (shaded area) weighted by a Gaussian distribution centred at 15 ns (see the text). Here, $T_{2,s}=120\mu \mathrm{s}$ and $T_{2,c}=400\mathrm{ns}$.

Figure 4

Concurrence C (measure of entanglement) as a function of charge decoherence for different levels of purification, without (top) and with (bottom) errors in the local quantum gates in the purification process. The dashed red line represents perfect unitary evolution (no decoherence) and perfect charge measurements and the only source of imperfections is the fact that our fast, resonant dynamics in Eq. (3) cannot exactly generate a Bell pair.

Figure 5

Entanglement purification protocol circuit. Here three rounds are displayed, but, depending on the noise model, two rounds might be sufficient. The first round addresses bit flip errors in the input noisy, raw Bell pairs, while the second round reduces phase flip errors. Then, in the third round we repeat the same procedure as in the first round but with two level-two Bell pairs as input. The fourth round (not shown here) reuses the circuit in the second round but with two level-three Bell pairs as input. If both bit flip and phase flip errors are present, the protocol should terminate at either the second or the fourth round depending on the noise rate. Errors in the purification process are accounted for by adding depolarizing noise after each local one- and two-qubit gate, and measurement errors ($99.9\mathrm{\%}$ fidelity) at each round (not displayed here). While the figure shows the canonical choice in entanglement purification, we tailored the protocol towards the specific error model of cavities by inserting further single-qubit rotations.

Figure 6

Distillation process flow chart for two rounds of distillation. The blue-shaded box represents the steps for one round of purification. It requires the generation of two raw Bell pairs (steps 1 and 2), a process that fails with probability $p_{\mathrm{raw}}\approx 0.10$. The distillation process (step 3) fails with probability $p_1\approx 0.08$. After having successfully generated two level-one Bell pairs, we perform a second round of distillation to get the final purified Bell pair, which fails with probability $p_2\approx 0.06$. The probabilities are given for the $T_{2,s}=120\mu \mathrm{s}$ and $T_{2,c}=400\mathrm{ns}$ case.

Figure 7

Energy level diagram of a silicon DQD containing spin, valley, and orbital degrees of freedom for $t_c=25\mu \mathrm{eV}$, $B=40\mu \mathrm{eV}$, $b_x=b_z=1\mu \mathrm{eV}$, ${\varphi }_L=\pi /6$, ${\varphi }_R=0$, $|{\mathrm{\Delta }}_L|=60\mu \mathrm{eV}$, and $|{\mathrm{\Delta }}_L|=70\mu \mathrm{eV}$. As the detuning is swept, an electron state traverses from left to right. Both adiabatic (black arrows) and diabatic (red arrows) transitions may occur, and the initial spin state will coherently evolve into a superposition of output states. Leakage to the excited orbital sector will result in a mixed spin-valley state.

Figure 8

Concurrence between two spins, initialized in a Bell state, as one is shuttled down an array of 25 quantum dots. The value associated with each dot $i$ corresponds to the concurrence within the ground valley subspace at the end of an $i$-quantum-dot chain. Valley splittings and magnetic fields are identical for all traces, while the effect of valley phase fluctuations and different bare tunnel couplings is studied. Valley phase differences are sampled from a normal distribution with mean 0 and (top) $\sigma =\pi /4$ and (bottom) $\sigma =\pi /2$. Note that increasing (decreasing) entanglement between the valley degree of freedom and the shuttled spin decreases (increases) the concurrence between the (initially maximally entangled) two spins due to the monogamy of entanglement. The plots show a typical instance of the effect of random phase differences as one would encounter in a given physical device, whereas the fidelity figures we show later are averaged over a large number of random instances.

Figure 9

Comparing output density matrices in the case of cavity-based and shuttling-based approaches. Absolute values of matrix elements in the Bell basis, assuming imperfection parameters $T_{2,c}=400\mathrm{ns}$ (top) and ${\mathrm{SD}}_{\varphi }=\pi /4$ (bottom). The matrix associated with shuttling before purification corresponds to an average over multiple instances of 25-dot chains and is given up to local rotations. We need two rounds for the cavity-based approach, and only one for the shuttling-based one. The largest fidelity with respect to the closest of the four canonical Bell states is increased from $94.5\mathrm{\%}$ to $99.5\mathrm{\%}$ for the cavity-based method, and from $95.7\mathrm{\%}$ to $99.6\mathrm{\%}$ for shuttling.

Figure 10

Circuit representation of the error mitigation technique: the quantum cores (Qcores) perform the same quantum computation independently in parallel. A derangement operation $D_n$ is then applied immediately prior to measurement that uses the copies to validate each other. Errors in estimating the observable $\sigma$ are suppressed exponentially when increasing the number $n$ of cores. As each core consists of $N$ computational qubits, we show that the derangement circuit can be implemented efficiently by distributing $N(n-1)$ Bell pairs between the quantum cores and performing $N(n-1)$ controlled-swap operations locally. This figure has been adapted from Ref. [26].

Figure 11

(a) Circuit for teleporting a single-qubit state from core $B$ to core $A$ by consuming a Bell pair (wavy line) that has been distributed between the two cores; (b) after teleporting the single qubit we can locally apply the derangement circuit in core $A$, i.e., the controlled-swap operation between the two single qubits. The lower qubit is reset and repeating this procedure for all computational qubits $k$ implements the entire derangement circuit. Imperfections in the Bell pairs have a formally equivalent effect to a depolarizing channel applied during the computation and therefore the derangement circuit mitigates these imperfections. (c) Numerical simulation of a spin-ring Hamiltonian. Error in estimating the ground-state energy as a function of the expected number of errors in the state-preparation circuit ($\xi$). The unmitigated errors (red squares) are impressively removed by the error suppression technique (assuming noiseless derangement circuits) using $n$ copies of the computational state (dashed lines). In the practically relevant regime when $0.1\le \xi \le 5$, the performance of the ESD-VD technique is similar to the ideal one even when taking into account imperfections in the derangement circuit (brown stars) or imperfections in the Bell pair distribution (black crosses) or both (magenta circles).

Figure 12

Evolution of the photon’s state after a successful charge measurement. When we stop the interaction and make the measurements, there is approximately a $0.96$ probability of finding the cavity in the vacuum state.

Figure 13

Concurrence C (measure of entanglement) as a function of charge decoherence for different levels of purification, without (left) and with (right) errors in the local quantum gates in the purification process. The dashed red line represents a perfect unitary evolution (no decoherence) and perfect charge measurements and the only source of imperfections is the fact that our fast, resonant dynamics in Eq. (3) cannot exactly generate a Bell pair.

Figure 14

Energy-level diagram of $H_{2e}$ from Eq. (B2) in the positive-detuning region where electrons only occupy the ground orbital wave function. Here, $U=1\mathrm{meV}$, $t=5\mu \mathrm{eV}$, $E_{Z,L}=51\mu \mathrm{eV}$, $E_{Z,R}=49\mu \mathrm{eV}$, $E_{V,L}=90\mu \mathrm{eV}$, and $E_{Z,R}=110\mu \mathrm{eV}$. In this experimentally realistic regime, $\mathrm{\Delta }{E}_Z>0>\mathrm{\Delta }{E}_V$ and $|\mathrm{\Delta }{E}_V|>|\mathrm{\Delta }{E}_Z|$, although the Zeeman difference here has been exaggerated to enhance the visibility of distinct traces. The state labels are ordered as ${|vs⟩}_L{|vs⟩}_R$. The highlighted labels indicate the four states where the right dot is initially populated with an ancilla ground-valley or ground-spin state, and the left dot is populated with a shuttled spin-valley state. Of these four states, only those with the left dot electron in the excited valley state (red) will adiabatically evolve to $(0,2)$, while the states in the ground valley (blue) will remain in $(1,1)$. Therefore, such a double-dot arrangement along with a proximal charge sensor can be used to perform a valley-to-charge projective measurement while preserving the spin state.

Figure 15

Left: Pauli transfer matrix acting on the shuttled spin, corresponding to the 25-dot shuttling channel $\mathcal{S}$ shown in (a) when ${\mathrm{SD}}_{\varphi }=\pi /4$ and $t_c=30\mu \mathrm{eV}$. Red bars indicate positive matrix elements, while blue bars indicate negative matrix elements. Middle: calculation of the unpurified and purified spin-spin concurrences when shuttling through a DQD with a bare tunnel coupling of $t_c=20\mu \mathrm{eV}$ and equivalent valley splittings of $E_V=2|\mathrm{\Delta }|=150\mu \mathrm{eV}$. Here $b_x=b_z=1\mu \mathrm{eV}$ and $B=40\mu \mathrm{eV}$. The tunnel coupling has been decreased in order to maximize the effect of charge noise. Right: the difference between noisy and noiseless simulations shows that charge noise has a negligible effect on the concurrence at the tunnel couplings and sweep speeds of interest.

Figure 16

Quantum teleportation by injecting a Bell pair other than the required $|{\mathrm{\Phi }}^{+}⟩$ is equivalent to applying a Pauli operator ${\sigma }_{\alpha }$ to the teleported qubit state as $\pm {\sigma }_{\alpha }|\psi ⟩$, where the Pauli operator maps between the two Bell states as $|{\mathrm{\Phi }}_{\alpha }⟩:={\sigma }_{\alpha }|{\mathrm{\Phi }}^{+}⟩$. Because of linearity, using an imperfect, incoherent mixture of input Bell states for teleportation is equivalent to applying a Pauli noise channel to the teleported single-qubit state. Additionally, applying twirling guarantees that this error channel is a single-qubit depolarizing channel.